Numerical Linear Algebra

Biostat/Biomath M257

Author

Dr. Hua Zhou @ UCLA

Published

April 16, 2024

System information (for reproducibility):

versioninfo()
Julia Version 1.10.2
Commit bd47eca2c8a (2024-03-01 10:14 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: macOS (arm64-apple-darwin22.4.0)
  CPU: 12 × Apple M2 Max
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, apple-m1)
Threads: 8 default, 0 interactive, 4 GC (on 8 virtual cores)
Environment:
  JULIA_NUM_THREADS = 8
  JULIA_EDITOR = code

Load packages:

using Pkg

Pkg.activate(pwd())
Pkg.instantiate()
Pkg.status()
  Activating project at `~/Documents/github.com/ucla-biostat-257/2024spring/slides/08-numalgintro`
Status `~/Documents/github.com/ucla-biostat-257/2024spring/slides/08-numalgintro/Project.toml`
  [6e4b80f9] BenchmarkTools v1.5.0
  [0e44f5e4] Hwloc v3.0.1
  [bdcacae8] LoopVectorization v0.12.169
  [6f49c342] RCall v0.14.1
  [37e2e46d] LinearAlgebra
  [9a3f8284] Random

1 Introduction

  • Topics in numerical algebra:

    • BLAS
    • solve linear equations \(\mathbf{A} \mathbf{x} = \mathbf{b}\)
    • regression computations \(\mathbf{X}^T \mathbf{X} \beta = \mathbf{X}^T \mathbf{y}\)
    • eigen-problems \(\mathbf{A} \mathbf{x} = \lambda \mathbf{x}\)
    • generalized eigen-problems \(\mathbf{A} \mathbf{x} = \lambda \mathbf{B} \mathbf{x}\)
    • singular value decompositions \(\mathbf{A} = \mathbf{U} \Sigma \mathbf{V}^T\)
    • iterative methods for numerical linear algebra
  • Except for the iterative methods, most of these numerical linear algebra tasks are implemented in the BLAS and LAPACK libraries. They form the building blocks of most statistical computing tasks (optimization, MCMC).

  • Our major goal (or learning objectives) is to

    1. know the complexity (flop count) of each task
    2. be familiar with the BLAS and LAPACK functions (what they do)
    3. do not re-invent wheels by implementing these dense linear algebra subroutines by yourself
    4. understand the need for iterative methods
    5. apply appropriate numerical algebra tools to various statistical problems
  • All high-level languages (Julia, Matlab, Python, R) call BLAS and LAPACK for numerical linear algebra.

    • Julia offers more flexibility by exposing interfaces to many BLAS/LAPACK subroutines directly. See documentation.

2 BLAS

  • BLAS stands for basic linear algebra subprograms.

  • See netlib for a complete list of standardized BLAS functions.

  • There are many implementations of BLAS.

    • Netlib provides a reference implementation.
    • Matlab uses Intel’s MKL (mathematical kernel libaries). MKL implementation is the gold standard on market. It is not open source but the compiled library is free for Linux and MacOS. However, not surprisingly, it only works on Intel CPUs.
    • Julia uses OpenBLAS. OpenBLAS is the best cross-platform, open source implementation. With the MKL.jl package, it’s also very easy to use MKL in Julia.
  • There are 3 levels of BLAS functions.

Level Example Operation Name Dimension Flops
1 \(\alpha \gets \mathbf{x}^T \mathbf{y}\) dot product \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^n\) \(2n\)
1 \(\mathbf{y} \gets \mathbf{y} + \alpha \mathbf{x}\) axpy \(\alpha \in \mathbb{R}\), \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^n\) \(2n\)
2 \(\mathbf{y} \gets \mathbf{y} + \mathbf{A} \mathbf{x}\) gaxpy \(\mathbf{A} \in \mathbb{R}^{m \times n}\), \(\mathbf{x} \in \mathbb{R}^n\), \(\mathbf{y} \in \mathbb{R}^m\) \(2mn\)
2 \(\mathbf{A} \gets \mathbf{A} + \mathbf{y} \mathbf{x}^T\) rank one update \(\mathbf{A} \in \mathbb{R}^{m \times n}\), \(\mathbf{x} \in \mathbb{R}^n\), \(\mathbf{y} \in \mathbb{R}^m\) \(2mn\)
3 \(\mathbf{C} \gets \mathbf{C} + \mathbf{A} \mathbf{B}\) matrix multiplication \(\mathbf{A} \in \mathbb{R}^{m \times p}\), \(\mathbf{B} \in \mathbb{R}^{p \times n}\), \(\mathbf{C} \in \mathbb{R}^{m \times n}\) \(2mnp\)
  • Typical BLAS functions support single precision (S), double precision (D), complex (C), and double complex (Z).

3 Examples

The form of a mathematical expression and the way the expression should be evaluated in actual practice may be quite different.

Some operations appear as level-3 but indeed are level-2.

Example 1. A common operation in statistics is column scaling or row scaling \[ \begin{eqnarray*} \mathbf{A} &=& \mathbf{A} \mathbf{D} \quad \text{(column scaling)} \\ \mathbf{A} &=& \mathbf{D} \mathbf{A} \quad \text{(row scaling)}, \end{eqnarray*} \] where \(\mathbf{D}\) is diagonal. For example, in generalized linear models (GLMs), the Fisher information matrix takes the form
\[ \mathbf{X}^T \mathbf{W} \mathbf{X}, \] where \(\mathbf{W}\) is a diagonal matrix with observation weights on diagonal.

Column and row scalings are essentially level-2 operations!

using BenchmarkTools, LinearAlgebra, Random

Random.seed!(257) # seed
n = 2000
A = rand(n, n) # n-by-n matrix
d = rand(n)  # n vector
D = Diagonal(d) # diagonal matrix with d as diagonal
2000×2000 Diagonal{Float64, Vector{Float64}}:
 0.0416032   ⋅         ⋅         ⋅       …   ⋅         ⋅         ⋅ 
  ⋅         0.887679   ⋅         ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅        0.102233   ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅        0.08407      ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅       …   ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅         ⋅ 
 ⋮                                       ⋱                      
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅       …   ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅          0.213471   ⋅         ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅        0.870533   ⋅ 
  ⋅          ⋅         ⋅         ⋅           ⋅         ⋅        0.318876
Dfull = diagm(d) # convert to full matrix
2000×2000 Matrix{Float64}:
 0.0416032  0.0       0.0       0.0      …  0.0       0.0       0.0
 0.0        0.887679  0.0       0.0         0.0       0.0       0.0
 0.0        0.0       0.102233  0.0         0.0       0.0       0.0
 0.0        0.0       0.0       0.08407     0.0       0.0       0.0
 0.0        0.0       0.0       0.0         0.0       0.0       0.0
 0.0        0.0       0.0       0.0      …  0.0       0.0       0.0
 0.0        0.0       0.0       0.0         0.0       0.0       0.0
 0.0        0.0       0.0       0.0         0.0       0.0       0.0
 0.0        0.0       0.0       0.0         0.0       0.0       0.0
 0.0        0.0       0.0       0.0         0.0       0.0       0.0
 ⋮                                       ⋱                      
 0.0        0.0       0.0       0.0         0.0       0.0       0.0
 0.0        0.0       0.0       0.0         0.0       0.0       0.0
 0.0        0.0       0.0       0.0         0.0       0.0       0.0
 0.0        0.0       0.0       0.0         0.0       0.0       0.0
 0.0        0.0       0.0       0.0      …  0.0       0.0       0.0
 0.0        0.0       0.0       0.0         0.0       0.0       0.0
 0.0        0.0       0.0       0.0         0.213471  0.0       0.0
 0.0        0.0       0.0       0.0         0.0       0.870533  0.0
 0.0        0.0       0.0       0.0         0.0       0.0       0.318876
# this is calling BLAS routine for matrix multiplication: O(n^3) flops
# this is SLOW!
@benchmark $A * $Dfull
BenchmarkTools.Trial: 96 samples with 1 evaluation.
 Range (minmax):  51.614 ms 57.005 ms   GC (min … max): 0.00% … 2.41%
 Time  (median):     51.760 ms                GC (median):    0.00%
 Time  (mean ± σ):   52.256 ms ± 866.879 μs   GC (mean ± σ):  0.86% ± 1.22%
   ▃█                                                           
  ▄██▄▁▂▁▁▁▁▁▁▁▁▁▁▁▂▁▁▁▁▁▁▃▅▇▅▄▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▁▂ ▁
  51.6 ms         Histogram: frequency by time         54.7 ms <
 Memory estimate: 30.52 MiB, allocs estimate: 2.
# dispatch to special method for diagonal matrix multiplication.
# columnwise scaling: O(n^2) flops
@benchmark $A * $D
BenchmarkTools.Trial: 1897 samples with 1 evaluation.
 Range (minmax):  2.127 ms  3.999 ms   GC (min … max):  0.00% … 22.40%
 Time  (median):     2.442 ms                GC (median):     0.00%
 Time  (mean ± σ):   2.632 ms ± 438.140 μs   GC (mean ± σ):  10.90% ± 12.67%
    ▆█▇▅▄                                                      
  ▃▆██████▇▆▇▆▆▆▆▆▅▅▄▄▄▃▃▁▂▂▂▁▂▂▂▂▁▁▁▂▄▅██▆▇▇▆▅▆▅▅▄▄▃▄▄▃▃▃▃ ▄
  2.13 ms         Histogram: frequency by time        3.51 ms <
 Memory estimate: 30.52 MiB, allocs estimate: 2.
# Or we can use broadcasting (with recycling)
@benchmark $A .* transpose($d)
BenchmarkTools.Trial: 1883 samples with 1 evaluation.
 Range (minmax):  2.126 ms  4.233 ms   GC (min … max):  0.00% … 27.57%
 Time  (median):     2.469 ms                GC (median):     0.00%
 Time  (mean ± σ):   2.652 ms ± 451.323 μs   GC (mean ± σ):  10.93% ± 12.70%
    ▅█▆▂                                                       
  ▃▆█████▇▆▆▅▅▇▅▅▆▆▄▃▃▃▂▂▂▂▂▂▂▂▂▂▁▂▄▇█▇█▆▅▄▅▄▄▄▄▃▄▄▃▂▂▂▂▁▂▂ ▄
  2.13 ms         Histogram: frequency by time        3.67 ms <
 Memory estimate: 30.52 MiB, allocs estimate: 2.
# in-place: avoid allocate space for result
# rmul!: compute matrix-matrix product A*B, overwriting A, and return the result.
@benchmark rmul!($A, $D)
BenchmarkTools.Trial: 3417 samples with 1 evaluation.
 Range (minmax):  1.263 ms 4.679 ms   GC (min … max): 0.00% … 0.00%
 Time  (median):     1.460 ms               GC (median):    0.00%
 Time  (mean ± σ):   1.461 ms ± 84.548 μs   GC (mean ± σ):  0.00% ± 0.00%
                        ▂▂▄▆▄▇▆▄▄▆▃▃▁▄▂▁▄▆▄▆▆█▆▄▃           
  ▂▂▁▂▃▃▃▃▄▃▃▄▄▄▅▄▆▆▆▆▇███████████████████████████▆▅▅▃▃▄▃▃ ▅
  1.26 ms        Histogram: frequency by time         1.6 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.
# In-place broadcasting 
@benchmark $A .= $A .* transpose($d)
BenchmarkTools.Trial: 3174 samples with 1 evaluation.
 Range (minmax):  1.423 ms 5.454 ms   GC (min … max): 0.00% … 0.00%
 Time  (median):     1.567 ms               GC (median):    0.00%
 Time  (mean ± σ):   1.573 ms ± 91.401 μs   GC (mean ± σ):  0.00% ± 0.00%
                             ▂▃▃█▆▇▄▃                      
  ▂▁▂▁▂▂▂▁▁▂▂▂▁▂▂▂▂▂▂▂▂▂▂▂▄▅▇█████████▆▅▅▃▃▃▃▃▃▃▃▃▃▂▃▃▃▂▂ ▄
  1.42 ms        Histogram: frequency by time        1.67 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.

Exercise: Try @turbo (SIMD) and @tturbo (multi-threaded SIMD) from LoopVectorization.jl package.

Note: In R or Matlab, diag(d) will create a full matrix. Be cautious using diag function: do we really need a full diagonal matrix?

using RCall

R"""
d <- runif(5)
diag(d)
"""
RObject{RealSxp}
          [,1]      [,2]      [,3]      [,4]      [,5]
[1,] 0.3485362 0.0000000 0.0000000 0.0000000 0.0000000
[2,] 0.0000000 0.5413159 0.0000000 0.0000000 0.0000000
[3,] 0.0000000 0.0000000 0.1638823 0.0000000 0.0000000
[4,] 0.0000000 0.0000000 0.0000000 0.5827122 0.0000000
[5,] 0.0000000 0.0000000 0.0000000 0.0000000 0.9024851
# This works only when Matlab is installed
using MATLAB

mat"""
d = rand(5, 1)
diag(d)
"""
ArgumentError: ArgumentError: Package MATLAB not found in current path.
- Run `import Pkg; Pkg.add("MATLAB")` to install the MATLAB package.

Example 2. Innter product between two matrices \(\mathbf{A}, \mathbf{B} \in \mathbb{R}^{m \times n}\) is often written as \[ \text{trace}(\mathbf{A}^T \mathbf{B}), \text{trace}(\mathbf{B} \mathbf{A}^T), \text{trace}(\mathbf{A} \mathbf{B}^T), \text{ or } \text{trace}(\mathbf{B}^T \mathbf{A}). \] They appear as level-3 operation (matrix multiplication with \(O(m^2n)\) or \(O(mn^2)\) flops).

Random.seed!(123)
n = 2000
A, B = randn(n, n), randn(n, n)

# slow way to evaluate tr(A'B): 2mn^2 flops
@benchmark tr(transpose($A) * $B)
BenchmarkTools.Trial: 95 samples with 1 evaluation.
 Range (minmax):  50.924 ms59.158 ms   GC (min … max): 0.00% … 3.94%
 Time  (median):     52.136 ms               GC (median):    0.00%
 Time  (mean ± σ):   52.486 ms ±  1.530 ms   GC (mean ± σ):  0.86% ± 1.46%
  █                                                           
  █▅▃▄▆▃▇▅███▃▆▅▁▃▃▄▁▄▅▄▄▃▁▃▁▃▁▃▄▁▁▁▁▁▃▁▁▁▁▁▁▁▁▁▁▁▁▁▁▃▁▁▁▁▃ ▁
  50.9 ms         Histogram: frequency by time        58.3 ms <
 Memory estimate: 30.52 MiB, allocs estimate: 2.

But \(\text{trace}(\mathbf{A}^T \mathbf{B}) = <\text{vec}(\mathbf{A}), \text{vec}(\mathbf{B})>\). The latter is level-1 BLAS operation with \(O(mn)\) flops.

# smarter way to evaluate tr(A'B): 2mn flops
@benchmark dot($A, $B)
BenchmarkTools.Trial: 2732 samples with 1 evaluation.
 Range (minmax):  1.649 ms 1.984 ms   GC (min … max): 0.00% … 0.00%
 Time  (median):     1.823 ms               GC (median):    0.00%
 Time  (mean ± σ):   1.827 ms ± 31.177 μs   GC (mean ± σ):  0.00% ± 0.00%
                               ▁▁▁▃▄▇█▄▄▁                    
  ▂▁▁▁▁▁▁▁▁▁▁▁▂▁▂▁▂▁▁▁▁▁▁▁▁▂▅▆███████████▇▆▅▄▄▃▃▃▃▃▃▃▃▃▃▃▂▃ ▃
  1.65 ms        Histogram: frequency by time        1.93 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.

Example 3. Similarly \(\text{diag}(\mathbf{A}^T \mathbf{B})\) can be calculated in \(O(mn)\) flops.

# slow way to evaluate diag(A'B): O(n^3)
@benchmark diag(transpose($A) * $B)
BenchmarkTools.Trial: 93 samples with 1 evaluation.
 Range (minmax):  50.993 ms100.033 ms   GC (min … max): 0.00% … 1.47%
 Time  (median):     52.402 ms                GC (median):    0.00%
 Time  (mean ± σ):   53.965 ms ±   6.802 ms   GC (mean ± σ):  0.90% ± 1.25%
  █▅▅▄                                                         
  ████▅▅▁▅▁▁▁▁▇▅▁▁▁▁▁▁▁▁▅▁▁▁▁▁▁▁▁▅▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▅ ▁
  51 ms         Histogram: log(frequency) by time      88.9 ms <
 Memory estimate: 30.53 MiB, allocs estimate: 3.
# smarter way to evaluate diag(A'B): O(n^2)
@benchmark Diagonal(vec(sum($A .* $B, dims = 1)))
BenchmarkTools.Trial: 1294 samples with 1 evaluation.
 Range (minmax):  3.305 ms  5.589 ms   GC (min … max): 0.00% … 23.28%
 Time  (median):     3.625 ms                GC (median):    0.00%
 Time  (mean ± σ):   3.860 ms ± 513.706 μs   GC (mean ± σ):  8.86% ± 10.65%
   ▄█▆▃                                                        
  ▆████████▇▇▆█▄█▅▅▃▄▂▂▂▂▁▁▁▁▁▁▁▁▁▁▁▂▁▂▄▇▇▇▇▇▅▆▆▅▅▄▅▄▄▃▃▃▂▂ ▄
  3.3 ms          Histogram: frequency by time        4.88 ms <
 Memory estimate: 30.53 MiB, allocs estimate: 5.

To get rid of allocation of intermediate arrays at all, we can just write a double loop or use dot function.

function diag_matmul!(d, A, B)
    m, n = size(A)
    @assert size(B) == (m, n) "A and B should have same size"
    fill!(d, 0)
    for j in 1:n, i in 1:m
        d[j] += A[i, j] * B[i, j]
    end
    Diagonal(d)
end

d = zeros(eltype(A), size(A, 2))
@benchmark diag_matmul!($d, $A, $B)
BenchmarkTools.Trial: 1479 samples with 1 evaluation.
 Range (minmax):  3.117 ms 3.586 ms   GC (min … max): 0.00% … 0.00%
 Time  (median):     3.372 ms               GC (median):    0.00%
 Time  (mean ± σ):   3.379 ms ± 41.213 μs   GC (mean ± σ):  0.00% ± 0.00%
                                 ▁ ▁ ▆▅█▇▂                  
  ▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▁▁▁▁▁▁▁▆█▇████████▇▆▆▄▄▄▄▄▄▄▃▄▃▃▃ ▃
  3.12 ms        Histogram: frequency by time         3.5 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.

Exercise: Try @turbo (SIMD) and @tturbo (multi-threaded SIMD) from LoopVectorization.jl package.

4 Memory hierarchy and level-3 fraction

Key to high performance is effective use of memory hierarchy. True on all architectures.

  • Flop count is not the sole determinant of algorithm efficiency. Another important factor is data movement through the memory hierarchy.

Source: https://cs.brown.edu/courses/csci1310/2020/assign/labs/lab4.html

  • In Julia, we can query the CPU topology by the Hwloc.jl package. For example, this laptop runs an Apple M2 Max chip with 4 efficiency cores and 8 performance cores.
using Hwloc

topology_graphical()
/------------------------------------------------------------------------------------------------------------------------------------------------------------------\
| Machine (2208MB total)                                                                                                                                           |
|                                                                                                                                                                  |
| /----------------------------------------------------------------------------------------------------------------------------------------\  /------------------\ |
| | Package L#0                                                                                                                            |  | CoProc opencl0d0 | |
| |                                                                                                                                        |  |                  | |
| | /------------------------------------------------------------------------------------------------------------------------------------\ |  | 38 compute units | |
| | | NUMANode L#0 P#0 (2208MB)                                                                                                          | |  |                  | |
| | \------------------------------------------------------------------------------------------------------------------------------------/ |  | 72 GB            | |
| |                                                                                                                                        |  \------------------/ |
| | /----------------------------------------------------------------\  /----------------------------------------------------------------\ |                       |
| | | L2 (4096KB)                                                    |  | L2 (16MB)                                                      | |                       |
| | \----------------------------------------------------------------/  \----------------------------------------------------------------/ |                       |
| |                                                                                                                                        |                       |
| | /-------------\  /-------------\  /-------------\  /-------------\  /-------------\  /-------------\  /-------------\  /-------------\ |                       |
| | | L1d (64KB)  |  | L1d (64KB)  |  | L1d (64KB)  |  | L1d (64KB)  |  | L1d (128KB) |  | L1d (128KB) |  | L1d (128KB) |  | L1d (128KB) | |                       |
| | \-------------/  \-------------/  \-------------/  \-------------/  \-------------/  \-------------/  \-------------/  \-------------/ |                       |
| |                                                                                                                                        |                       |
| | /-------------\  /-------------\  /-------------\  /-------------\  /-------------\  /-------------\  /-------------\  /-------------\ |                       |
| | | L1i (128KB) |  | L1i (128KB) |  | L1i (128KB) |  | L1i (128KB) |  | L1i (192KB) |  | L1i (192KB) |  | L1i (192KB) |  | L1i (192KB) | |                       |
| | \-------------/  \-------------/  \-------------/  \-------------/  \-------------/  \-------------/  \-------------/  \-------------/ |                       |
| |                                                                                                                                        |                       |
| | /-------------\  /-------------\  /-------------\  /-------------\  /-------------\  /-------------\  /-------------\  /-------------\ |                       |
| | | Core L#0    |  | Core L#1    |  | Core L#2    |  | Core L#3    |  | Core L#4    |  | Core L#5    |  | Core L#6    |  | Core L#7    | |                       |
| | |             |  |             |  |             |  |             |  |             |  |             |  |             |  |             | |                       |
| | | /---------\ |  | /---------\ |  | /---------\ |  | /---------\ |  | /---------\ |  | /---------\ |  | /---------\ |  | /---------\ | |                       |
| | | | PU L#0  | |  | | PU L#1  | |  | | PU L#2  | |  | | PU L#3  | |  | | PU L#4  | |  | | PU L#5  | |  | | PU L#6  | |  | | PU L#7  | | |                       |
| | | |         | |  | |         | |  | |         | |  | |         | |  | |         | |  | |         | |  | |         | |  | |         | | |                       |
| | | |   P#0   | |  | |   P#1   | |  | |   P#2   | |  | |   P#3   | |  | |   P#4   | |  | |   P#5   | |  | |   P#6   | |  | |   P#7   | | |                       |
| | | \---------/ |  | \---------/ |  | \---------/ |  | \---------/ |  | \---------/ |  | \---------/ |  | \---------/ |  | \---------/ | |                       |
| | \-------------/  \-------------/  \-------------/  \-------------/  \-------------/  \-------------/  \-------------/  \-------------/ |                       |
| |                                                                                                                                        |                       |
| | /----------------------------------------------------------------\                                                                     |                       |
| | | L2 (16MB)                                                      |                                                                     |                       |
| | \----------------------------------------------------------------/                                                                     |                       |
| |                                                                                                                                        |                       |
| | /-------------\  /-------------\  /-------------\  /-------------\                                                                     |                       |
| | | L1d (128KB) |  | L1d (128KB) |  | L1d (128KB) |  | L1d (128KB) |                                                                     |                       |
| | \-------------/  \-------------/  \-------------/  \-------------/                                                                     |                       |
| |                                                                                                                                        |                       |
| | /-------------\  /-------------\  /-------------\  /-------------\                                                                     |                       |
| | | L1i (192KB) |  | L1i (192KB) |  | L1i (192KB) |  | L1i (192KB) |                                                                     |                       |
| | \-------------/  \-------------/  \-------------/  \-------------/                                                                     |                       |
| |                                                                                                                                        |                       |
| | /-------------\  /-------------\  /-------------\  /-------------\                                                                     |                       |
| | | Core L#8    |  | Core L#9    |  | Core L#10   |  | Core L#11   |                                                                     |                       |
| | |             |  |             |  |             |  |             |                                                                     |                       |
| | | /---------\ |  | /---------\ |  | /---------\ |  | /---------\ |                                                                     |                       |
| | | | PU L#8  | |  | | PU L#9  | |  | | PU L#10 | |  | | PU L#11 | |                                                                     |                       |
| | | |         | |  | |         | |  | |         | |  | |         | |                                                                     |                       |
| | | |   P#8   | |  | |   P#9   | |  | |  P#10   | |  | |  P#11   | |                                                                     |                       |
| | | \---------/ |  | \---------/ |  | \---------/ |  | \---------/ |                                                                     |                       |
| | \-------------/  \-------------/  \-------------/  \-------------/                                                                     |                       |
| \----------------------------------------------------------------------------------------------------------------------------------------/                       |
\------------------------------------------------------------------------------------------------------------------------------------------------------------------/
  • For example, Xeon X5650 CPU has a theoretical throughput of 128 DP GFLOPS but a max memory bandwidth of 32GB/s.

  • Can we keep CPU cores busy with enough deliveries of matrix data and ship the results to memory fast enough to avoid backlog?
    Answer: use high-level BLAS as much as possible.

BLAS Dimension Mem. Refs. Flops Ratio
Level 1: \(\mathbf{y} \gets \mathbf{y} + \alpha \mathbf{x}\) \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^n\) \(3n\) \(2n\) 3:2
Level 2: \(\mathbf{y} \gets \mathbf{y} + \mathbf{A} \mathbf{x}\) \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^n\), \(\mathbf{A} \in \mathbb{R}^{n \times n}\) \(n^2\) \(2n^2\) 1:2
Level 3: \(\mathbf{C} \gets \mathbf{C} + \mathbf{A} \mathbf{B}\) \(\mathbf{A}, \mathbf{B}, \mathbf{C} \in\mathbb{R}^{n \times n}\) \(4n^2\) \(2n^3\) 2:n
  • Higher level BLAS (3 or 2) make more effective use of arithmetic logic units (ALU) by keeping them busy. Surface-to-volume effect.

Source: Jack Dongarra’s slides.

  • A distinction between LAPACK and LINPACK (older version of R uses LINPACK) is that LAPACK makes use of higher level BLAS as much as possible (usually by smart partitioning) to increase the so-called level-3 fraction.

  • To appreciate the efforts in an optimized BLAS implementation such as OpenBLAS (evolved from GotoBLAS), see the Quora question, especially the video. Bottomline is

Get familiar with (good implementations of) BLAS/LAPACK and use them as much as possible.

5 Effect of data layout

  • Data layout in memory affects algorithmic efficiency too. It is much faster to move chunks of data in memory than retrieving/writing scattered data.

  • Storage mode: column-major (Fortran, Matlab, R, Julia) vs row-major (C/C++).

  • Cache line is the minimum amount of cache which can be loaded and stored to memory.

    • x86 CPUs: 64 bytes
    • ARM CPUs: 32 bytes

  • In Julia, we can query the cache line size by Hwloc.jl.
# Apple Silicon (M1/M2 chips) don't have L3 cache
Hwloc.cachelinesize()
ErrorException: Your system doesn't seem to have an L3 cache.
  • Accessing column-major stored matrix by rows (\(ij\) looping) causes lots of cache misses.

  • Take matrix multiplication as an example \[ \mathbf{C} \gets \mathbf{C} + \mathbf{A} \mathbf{B}, \quad \mathbf{A} \in \mathbb{R}^{m \times p}, \mathbf{B} \in \mathbb{R}^{p \times n}, \mathbf{C} \in \mathbb{R}^{m \times n}. \] Assume the storage is column-major, such as in Julia. There are 6 variants of the algorithms according to the order in the triple loops.

    • jki or kji looping:
# inner most loop
for i in 1:m
    C[i, j] = C[i, j] + A[i, k] * B[k, j]
end
- `ikj` or `kij` looping:
# inner most loop        
for j in 1:n
    C[i, j] = C[i, j] + A[i, k] * B[k, j]
end
  • ijk or jik looping:
# inner most loop        
for k in 1:p
    C[i, j] = C[i, j] + A[i, k] * B[k, j]
end
  • We pay attention to the innermost loop, where the vector calculation occurs. The associated stride when accessing the three matrices in memory (assuming column-major storage) is
Variant A Stride B Stride C Stride
\(jki\) or \(kji\) Unit 0 Unit
\(ikj\) or \(kij\) 0 Non-Unit Non-Unit
\(ijk\) or \(jik\) Non-Unit Unit 0

Apparently the variants \(jki\) or \(kji\) are preferred.

"""
    matmul_by_loop!(A, B, C, order)

Overwrite `C` by `A * B`. `order` indicates the looping order for triple loop.
"""
function matmul_by_loop!(A::Matrix, B::Matrix, C::Matrix, order::String)
    
    m = size(A, 1)
    p = size(A, 2)
    n = size(B, 2)
    fill!(C, 0)
    
    if order == "jki"
        @inbounds for j = 1:n, k = 1:p, i = 1:m
            C[i, j] += A[i, k] * B[k, j]
        end
    end

    if order == "kji"
        @inbounds for k = 1:p, j = 1:n, i = 1:m
            C[i, j] += A[i, k] * B[k, j]
        end
    end
    
    if order == "ikj"
        @inbounds for i = 1:m, k = 1:p, j = 1:n
            C[i, j] += A[i, k] * B[k, j]
        end
    end

    if order == "kij"
        @inbounds for k = 1:p, i = 1:m, j = 1:n
            C[i, j] += A[i, k] * B[k, j]
        end
    end
    
    if order == "ijk"
        @inbounds for i = 1:m, j = 1:n, k = 1:p
            C[i, j] += A[i, k] * B[k, j]
        end
    end
    
    if order == "jik"
        @inbounds for j = 1:n, i = 1:m, k = 1:p
            C[i, j] += A[i, k] * B[k, j]
        end
    end
    
end

using Random

Random.seed!(123)
m, p, n = 2000, 100, 2000
A = rand(m, p)
B = rand(p, n)
C = zeros(m, n);
  • \(jki\) and \(kji\) looping:
using BenchmarkTools

@benchmark matmul_by_loop!($A, $B, $C, "jki")
BenchmarkTools.Trial: 86 samples with 1 evaluation.
 Range (minmax):  58.056 ms 59.729 ms   GC (min … max): 0.00% … 0.00%
 Time  (median):     58.433 ms                GC (median):    0.00%
 Time  (mean ± σ):   58.453 ms ± 234.579 μs   GC (mean ± σ):  0.00% ± 0.00%
           ▁ ▁█▂▁ ▅▁▂                                          
  ▃▁▁▃▆▁▅▃▆█▆███████▅▅▁▅▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▃▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▃ ▁
  58.1 ms         Histogram: frequency by time         59.6 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.
@benchmark matmul_by_loop!($A, $B, $C, "kji")
BenchmarkTools.Trial: 27 samples with 1 evaluation.
 Range (minmax):  185.771 ms190.199 ms   GC (min … max): 0.00% … 0.00%
 Time  (median):     186.726 ms                GC (median):    0.00%
 Time  (mean ± σ):   187.133 ms ±   1.237 ms   GC (mean ± σ):  0.00% ± 0.00%
      ▃█  █              ▃                                 
  ▇▇▇▁██▇▁█▁▁▁▇█▇▁▇▁▁▁▁▁▁▁▁▁▁█▇▁▇▁▁▁▇▁▁▁▁▇▁▁▇▁▁▁▁▁▁▁▁▇▁▁▁▁▁▁▇ ▁
  186 ms           Histogram: frequency by time          190 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.
  • \(ikj\) and \(kij\) looping:
@benchmark matmul_by_loop!($A, $B, $C, "ikj")
BenchmarkTools.Trial: 10 samples with 1 evaluation.
 Range (minmax):  512.687 ms522.756 ms   GC (min … max): 0.00% … 0.00%
 Time  (median):     515.130 ms                GC (median):    0.00%
 Time  (mean ± σ):   515.765 ms ±   2.925 ms   GC (mean ± σ):  0.00% ± 0.00%
  ▁▁        ▁▁ █            ▁                          ▁  
  ██▁▁▁▁▁▁▁▁██▁▁▁▁█▁█▁▁▁▁▁▁▁▁▁▁▁▁█▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█ ▁
  513 ms           Histogram: frequency by time          523 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.
@benchmark matmul_by_loop!($A, $B, $C, "kij")
BenchmarkTools.Trial: 10 samples with 1 evaluation.
 Range (minmax):  508.427 ms513.273 ms   GC (min … max): 0.00% … 0.00%
 Time  (median):     509.480 ms                GC (median):    0.00%
 Time  (mean ± σ):   510.326 ms ±   2.027 ms   GC (mean ± σ):  0.00% ± 0.00%
  ▁ █▁   ▁    ▁                             ▁▁ ▁  
  █▁██▁▁▁█▁▁▁▁▁▁▁▁▁█▁▁▁▁▁▁▁█▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁██▁█ ▁
  508 ms           Histogram: frequency by time          513 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.
  • \(ijk\) and \(jik\) looping:
@benchmark matmul_by_loop!($A, $B, $C, "ijk")
BenchmarkTools.Trial: 20 samples with 1 evaluation.
 Range (minmax):  247.799 ms257.115 ms   GC (min … max): 0.00% … 0.00%
 Time  (median):     249.820 ms                GC (median):    0.00%
 Time  (mean ± σ):   250.799 ms ±   2.507 ms   GC (mean ± σ):  0.00% ± 0.00%
  ▁ ▁    ▁▁▁█▁ ▁█ ▁    ▁           █       ▁       ▁         ▁  
  █▁█▁▁▁▁█████▁██▁█▁▁▁█▁▁▁▁▁▁▁▁▁▁▁█▁▁▁▁▁▁▁█▁▁▁▁▁▁▁█▁▁▁▁▁▁▁▁▁█ ▁
  248 ms           Histogram: frequency by time          257 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.
@benchmark matmul_by_loop!($A, $B, $C, "ijk")
BenchmarkTools.Trial: 21 samples with 1 evaluation.
 Range (minmax):  246.205 ms256.408 ms   GC (min … max): 0.00% … 0.00%
 Time  (median):     249.057 ms                GC (median):    0.00%
 Time  (mean ± σ):   249.382 ms ±   2.296 ms   GC (mean ± σ):  0.00% ± 0.00%
         ▃      ▃ ▃   █                                         
  ▇▁▇▁▁▇▁█▁▁▇▁▁▁█▁█▇█▁▁▁▁▁▁▁▇▁▁▁▇▁▁▁▁▁▁▇▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▇ ▁
  246 ms           Histogram: frequency by time          256 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.
  • Question: Can our loop beat BLAS? Julia wraps BLAS library for matrix multiplication. We see BLAS library wins hands down (multi-threading, Strassen algorithm, higher level-3 fraction by block outer product).
@benchmark mul!($C, $A, $B)
BenchmarkTools.Trial: 1882 samples with 1 evaluation.
 Range (minmax):  2.553 ms  9.490 ms   GC (min … max): 0.00% … 0.00%
 Time  (median):     2.619 ms                GC (median):    0.00%
 Time  (mean ± σ):   2.655 ms ± 259.992 μs   GC (mean ± σ):  0.00% ± 0.00%
       ▁▆▇▄  ▁▅▅▃            ▁▇█▆▂▁                            
  ▃▂▂▃▃██████████▇▅▄▃▃▃▃▃▃▄██████▅▄▄▃▃▃▃▂▂▂▃▂▂▂▁▂▂▂▂▂▁▂▂▂▁▂ ▄
  2.55 ms         Histogram: frequency by time         2.8 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.
# direct call of BLAS wrapper function
@benchmark LinearAlgebra.BLAS.gemm!('N', 'N', 1.0, $A, $B, 0.0, $C)
BenchmarkTools.Trial: 1903 samples with 1 evaluation.
 Range (minmax):  2.539 ms  7.890 ms   GC (min … max): 0.00% … 0.00%
 Time  (median):     2.565 ms                GC (median):    0.00%
 Time  (mean ± σ):   2.626 ms ± 245.462 μs   GC (mean ± σ):  0.00% ± 0.00%
   ▇█▄▃                                                        
  ▄████▅▄▃▃▂▂▂▂▂▂▂▂▂▂▂▂▂▃▄▅▆▅▅▅▅▆▄▄▃▃▃▂▂▂▂▃▂▂▂▂▁▂▂▁▂▂▂▂▂▂▂▂ ▃
  2.54 ms         Histogram: frequency by time        2.84 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.

Question (again): Can our loop beat BLAS?

Exercise: Annotate the loop in matmul_by_loop! by @turbo and @tturbo (multi-threading) and benchmark again.

6 BLAS in R

  • Tip for R users. Standard R distribution from CRAN uses a very out-dated BLAS/LAPACK library.
using RCall

R"""
sessionInfo()
"""
RObject{VecSxp}
R version 4.3.2 (2023-10-31)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Sonoma 14.4.1

Matrix products: default
BLAS:   /System/Library/Frameworks/Accelerate.framework/Versions/A/Frameworks/vecLib.framework/Versions/A/libBLAS.dylib 
LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.11.0

locale:
[1] C

time zone: America/Los_Angeles
tzcode source: internal

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

loaded via a namespace (and not attached):
[1] compiler_4.3.2
R"""
library(dplyr)
library(bench)
A <- $A
B <- $B
bench::mark(A %*% B) %>%
  print(width = Inf)
""";
┌ Warning: RCall.jl: 
│ Attaching package: 'dplyr'
│ 
│ The following objects are masked from 'package:stats':
│ 
│     filter, lag
│ 
│ The following objects are masked from 'package:base':
│ 
│     intersect, setdiff, setequal, union
│ 
└ @ RCall /Users/huazhou/.julia/packages/RCall/dDAVd/src/io.jl:172
┌ Warning: RCall.jl: Warning: Some expressions had a GC in every iteration; so filtering is disabled.
└ @ RCall /Users/huazhou/.julia/packages/RCall/dDAVd/src/io.jl:172
# A tibble: 1 x 13
  expression      min   median `itr/sec` mem_alloc `gc/sec` n_itr  n_gc
  <bch:expr> <bch:tm> <bch:tm>     <dbl> <bch:byt>    <dbl> <int> <dbl>
1 A %*% B       125ms    126ms      7.80    30.5MB     7.80     4     4
  total_time result                memory             time          
    <bch:tm> <list>                <list>             <list>        
1      513ms <dbl [2,000 x 2,000]> <Rprofmem [1 x 3]> <bench_tm [4]>
  gc              
  <list>          
1 <tibble [4 x 3]>
  • Re-build R from source using OpenBLAS or MKL will immediately boost linear algebra performance in R. Google build R using MKL to get started. Similarly we can build Julia using MKL.

  • Matlab uses MKL. Usually it’s very hard to beat Matlab in terms of linear algebra.

using MATLAB

mat"""
f = @() $A * $B;
timeit(f)
"""
ArgumentError: ArgumentError: Package MATLAB not found in current path.
- Run `import Pkg; Pkg.add("MATLAB")` to install the MATLAB package.

7 Avoid memory allocation: some examples

7.1 Transposing matrix is an expensive memory operation

In R, the command

t(A) %*% x

will first transpose A then perform matrix multiplication, causing unnecessary memory allocation

using Random, LinearAlgebra, BenchmarkTools
Random.seed!(123)

n = 1000
A = rand(n, n)
x = rand(n);
R"""
A <- $A
x <- $x
bench::mark(t(A) %*% x) %>%
  print(width = Inf)
""";
# A tibble: 1 x 13
  expression      min   median `itr/sec` mem_alloc `gc/sec` n_itr  n_gc
  <bch:expr> <bch:tm> <bch:tm>     <dbl> <bch:byt>    <dbl> <int> <dbl>
1 t(A) %*% x   1.76ms   2.11ms      472.    7.64MB     95.5   178    36
  total_time result            memory             time            
    <bch:tm> <list>            <list>             <list>          
1      377ms <dbl [1,000 x 1]> <Rprofmem [2 x 3]> <bench_tm [214]>
  gc                
  <list>            
1 <tibble [214 x 3]>

Julia is avoids transposing matrix whenever possible. The Transpose type only creates a view of the transpose of matrix data.

typeof(transpose(A))
Transpose{Float64, Matrix{Float64}}
fieldnames(typeof(transpose(A)))
(:parent,)
# same data in tranpose(A) and original matrix A
pointer(transpose(A).parent), pointer(A)
(Ptr{Float64} @0x0000000159ba0000, Ptr{Float64} @0x0000000159ba0000)
# dispatch to BLAS
# does *not* actually transpose the matrix
@benchmark transpose($A) * $x
BenchmarkTools.Trial: 10000 samples with 1 evaluation.
 Range (minmax):  22.500 μs77.792 μs   GC (min … max): 0.00% … 0.00%
 Time  (median):     23.500 μs               GC (median):    0.00%
 Time  (mean ± σ):   24.330 μs ±  3.956 μs   GC (mean ± σ):  0.00% ± 0.00%
  █▆▁ ▁▁▂▂▂▂▁                                            ▁ ▂
  █████████████▇▇▇▇███▇▇▇▆▅▅▅▆▆▅▆▅▅▄▄▃▃▁▁▁▃▁▁▁▁▁▁▁▁▁▁▁▁▆▇█ █
  22.5 μs      Histogram: log(frequency) by time      48.4 μs <
 Memory estimate: 8.00 KiB, allocs estimate: 1.
# pre-allocate result
out = zeros(size(A, 2))
@benchmark mul!($out, transpose($A), $x)
BenchmarkTools.Trial: 10000 samples with 1 evaluation.
 Range (minmax):  22.625 μs 2.375 ms   GC (min … max): 0.00% … 0.00%
 Time  (median):     22.916 μs               GC (median):    0.00%
 Time  (mean ± σ):   24.165 μs ± 29.290 μs   GC (mean ± σ):  0.00% ± 0.00%
  █      ▁                                                   ▁
  █▇▅████▇▇▇▆▆▇▇▆▆▇▇▅▅▄▃▄▄▅▅▅▃▃▃▃▁▁▁▁▃▁▄▁▁▁▁▁▁▃▁▁▁▁▁▁▁▁▃▄▇ █
  22.6 μs      Histogram: log(frequency) by time      49.4 μs <
 Memory estimate: 0 bytes, allocs estimate: 0.
# or call BLAS wrapper directly
@benchmark LinearAlgebra.BLAS.gemv!('T', 1.0, $A, $x, 0.0, $out)
BenchmarkTools.Trial: 10000 samples with 1 evaluation.
 Range (minmax):  22.542 μs79.375 μs   GC (min … max): 0.00% … 0.00%
 Time  (median):     22.792 μs               GC (median):    0.00%
 Time  (mean ± σ):   23.348 μs ±  2.238 μs   GC (mean ± σ):  0.00% ± 0.00%
  ▇                   ▁                                   ▂
  ███▆▁▅▆▅▅▅▅▄▄▅▇█▅▅▅▇▇██████▇▅▇▇▇▆▆▅▄▅▅▅▄▅▅▆▆▇▆▆▆▆▅▅▅▆▇▇▇▇ █
  22.5 μs      Histogram: log(frequency) by time      32.5 μs <
 Memory estimate: 0 bytes, allocs estimate: 0.

7.2 Broadcast (dot operation) fuses loops and avoids memory allocation

Broadcasting in Julia achieves vectorized code without creating intermediate arrays.

Suppose we want to calculate elementsize maximum of absolute values of two large arrays. In R or Matlab, the command

max(abs(X), abs(Y))

will create two intermediate arrays and then one result array.

using RCall

Random.seed!(123)
X, Y = rand(1000, 1000), rand(1000, 1000)

R"""
library(dplyr)
library(bench)
bench::mark(max(abs($X), abs($Y))) %>%
  print(width = Inf)
""";
# A tibble: 1 x 13
  expression                           min   median `itr/sec` mem_alloc `gc/sec`
  <bch:expr>                      <bch:tm> <bch:tm>     <dbl> <bch:byt>    <dbl>
1 max(abs(`#JL`$X), abs(`#JL`$Y))   3.44ms   3.72ms      268.    15.3MB     219.
  n_itr  n_gc total_time result    memory             time            
  <int> <dbl>   <bch:tm> <list>    <list>             <list>          
1    65    53      242ms <dbl [1]> <Rprofmem [2 x 3]> <bench_tm [118]>
  gc                
  <list>            
1 <tibble [118 x 3]>

In Julia, dot operations are fused so no intermediate arrays are created.

# no intermediate arrays created, only result array created
@benchmark max.(abs.($X), abs.($Y))
BenchmarkTools.Trial: 6090 samples with 1 evaluation.
 Range (minmax):  254.125 μs  3.850 ms   GC (min … max):  0.00% … 77.90%
 Time  (median):     696.666 μs                GC (median):     0.00%
 Time  (mean ± σ):   819.485 μs ± 416.072 μs   GC (mean ± σ):  15.00% ± 18.68%
  ▃▁         ▇█▆▄▃▃▂                            ▁      ▁▁▁    ▂
  ██▆▄▄▁▁▁▁▁████████▇▆▃▄▃▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▃▄▆▇███████▆██▇██████ █
  254 μs        Histogram: log(frequency) by time       2.32 ms <
 Memory estimate: 7.63 MiB, allocs estimate: 2.

Pre-allocating result array gets rid of memory allocation at all.

# no memory allocation at all!
Z = zeros(size(X)) # zero matrix of same size as X
@benchmark $Z .= max.(abs.($X), abs.($Y)) # .= (vs =) is important!
BenchmarkTools.Trial: 10000 samples with 1 evaluation.
 Range (minmax):  213.292 μs415.042 μs   GC (min … max): 0.00% … 0.00%
 Time  (median):     222.750 μs                GC (median):    0.00%
 Time  (mean ± σ):   230.535 μs ±  14.789 μs   GC (mean ± σ):  0.00% ± 0.00%
      ▆▃▃▆                   █▃                                  
  ▂▄▄█████▅▄▃▃▂▂▂▂▂▁▁▁▁▁▁▁▃██▃▄▄▃▂▂▂▂▂▂▂▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ ▂
  213 μs           Histogram: frequency by time          276 μs <
 Memory estimate: 0 bytes, allocs estimate: 0.

Exercise: Annotate the broadcasting by @turbo and @tturbo (multi-threading) and benchmark again.

7.3 Views

View avoids creating extra copy of matrix data.

Random.seed!(123)
A = randn(1000, 1000)

# sum entries in a sub-matrix
@benchmark sum($A[1:2:500, 1:2:500])
BenchmarkTools.Trial: 10000 samples with 1 evaluation.
 Range (minmax):  45.292 μs  3.645 ms   GC (min … max):  0.00% … 97.74%
 Time  (median):     68.708 μs                GC (median):     0.00%
 Time  (mean ± σ):   80.365 μs ± 156.568 μs   GC (mean ± σ):  14.19% ±  7.15%
                           █                                    
  ▄▃▂▁▂▂▃▃▂▂▂▂▂▂▂▂▂▂▂▂▁▁▂▂▇█▇▄▅▇▆▆▅▅▅▅▅▄▄▃▃▃▃▃▂▂▂▂▂▂▂▂▂▂▂▂▂▂ ▃
  45.3 μs         Histogram: frequency by time         91.6 μs <
 Memory estimate: 488.42 KiB, allocs estimate: 2.
# view avoids creating a separate sub-matrix
@benchmark sum(@view $A[1:2:500, 1:2:500])
BenchmarkTools.Trial: 10000 samples with 1 evaluation.
 Range (minmax):  50.708 μs106.583 μs   GC (min … max): 0.00% … 0.00%
 Time  (median):     55.042 μs                GC (median):    0.00%
 Time  (mean ± σ):   56.216 μs ±   3.059 μs   GC (mean ± σ):  0.00% ± 0.00%
                       ▂▂▁▁▁▁              ▁               ▁
  ▇▄▁▃▁▁▃▁▁▁▃▁▁▄█▇▇▆▆▅▆▆▆██████▇▇▆▆▆▆██████████▆▆▆▆▇▇▆▆▆▆▇▅▆ █
  50.7 μs       Histogram: log(frequency) by time      68.9 μs <
 Memory estimate: 0 bytes, allocs estimate: 0.

The @views macro, which can be useful in some operations.

@benchmark @views sum($A[1:2:500, 1:2:500])
BenchmarkTools.Trial: 10000 samples with 1 evaluation.
 Range (minmax):  54.917 μs106.917 μs   GC (min … max): 0.00% … 0.00%
 Time  (median):     55.042 μs                GC (median):    0.00%
 Time  (mean ± σ):   56.096 μs ±   2.914 μs   GC (mean ± σ):  0.00% ± 0.00%
  █             ▂▁ ▁ ▁                                        ▁
  █▅▄▇▄▃▄▅▃▄▃▄▃██▇███▇█▆▆▅▅▃▄▅▄▆▆▆▆▇▆▇▇▅▆█▇▇▆▇▅▄▄▄▄▄▅▅▅▅▅▅▅▅ █
  54.9 μs       Histogram: log(frequency) by time      68.2 μs <
 Memory estimate: 0 bytes, allocs estimate: 0.